I can see how this gets you E(valuei|comparisons) for each each item i, but not P((valuei)i∈items|comparisons). One of the advantages Ozzie raises is the possibility to keep track of correlations in value estimates, which requires more than the marginal expectations.
I’m not sure what you mean. I’m thinking about pairwise comparisons in the following way.
(a) Every pair of items i,j have a true ratio of expectations E(Xi)/E(Xj)=μij. I hope this is uncontroversial.
(b) We observe the variables Rij according to logRij∼logμij+ϵij for some some normally distributed ϵij. Error terms might be dependent, but that complicates the analysis. (And is most likely not worth it.) This step could be more controversial, as there are other possible models to use.
Note that you will get a distribution over every E(Xi) too with this approach, but that would be in the Bayesian sense, i.e., p(E(Xi)∣comparisons), when we have a prior over E(Xi).
I can see how this gets you E(valuei|comparisons) for each each item i, but not P((valuei)i∈items|comparisons). One of the advantages Ozzie raises is the possibility to keep track of correlations in value estimates, which requires more than the marginal expectations.
I’m not sure what you mean. I’m thinking about pairwise comparisons in the following way.
(a) Every pair of items i,j have a true ratio of expectations E(Xi)/E(Xj)=μij. I hope this is uncontroversial. (b) We observe the variables Rij according to logRij∼logμij+ϵij for some some normally distributed ϵij. Error terms might be dependent, but that complicates the analysis. (And is most likely not worth it.) This step could be more controversial, as there are other possible models to use.
Note that you will get a distribution over every E(Xi) too with this approach, but that would be in the Bayesian sense, i.e., p(E(Xi)∣comparisons), when we have a prior over E(Xi).